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A373402 Numbers k such that the k-th maximal antirun of prime numbers > 3 has length different from all prior maximal antiruns. Sorted list of positions of first appearances in A027833.

%I A373402 #8 Jun 10 2024 23:35:51
%S A373402 1,2,4,6,8,10,21,24,30,35,40,41,46,50,69,82,131,140,185,192,199,210,
%T A373402 248,251,271,277,325,406,423,458,645,748,811,815,826,831,987,1053,
%U A373402 1109,1426,1456,1590,1629,1870,1967,2060,2371,2607,2920,2946,3564,3681,4119
%N A373402 Numbers k such that the k-th maximal antirun of prime numbers > 3 has length different from all prior maximal antiruns. Sorted list of positions of first appearances in A027833.
%C A373402 The unsorted version is A373401.
%C A373402 For this sequence, we define an antirun to be an interval of positions at which consecutive primes differ by at least 3.
%H A373402 Gus Wiseman, <a href="/A373403/a373403.txt">Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers</a>.
%e A373402 The maximal antiruns of prime numbers > 3 begin:
%e A373402     5
%e A373402     7  11
%e A373402    13  17
%e A373402    19  23  29
%e A373402    31  37  41
%e A373402    43  47  53  59
%e A373402    61  67  71
%e A373402    73  79  83  89  97 101
%e A373402   103 107
%e A373402   109 113 127 131 137
%e A373402   139 149
%e A373402   151 157 163 167 173 179
%e A373402 The a(n)-th rows begin:
%e A373402     5
%e A373402     7  11
%e A373402    19  23  29
%e A373402    43  47  53  59
%e A373402    73  79  83  89  97 101
%e A373402   109 113 127 131 137
%t A373402 t=Length/@Split[Select[Range[4,10000],PrimeQ],#1+2!=#2&]//Most;
%t A373402 Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]
%Y A373402 For squarefree runs we have the triple (1,3,5), firsts of A120992.
%Y A373402 For prime runs we have the triple (1,2,3), firsts of A175632.
%Y A373402 For nonsquarefree runs we have A373199 (assuming sorted), firsts of A053797.
%Y A373402 For squarefree antiruns: A373200, unsorted A373128, firsts of A373127.
%Y A373402 For composite runs we have A373400, unsorted A073051.
%Y A373402 The unsorted version is A373401, firsts of A027833.
%Y A373402 For composite antiruns we have the triple (1,2,7), firsts of A373403.
%Y A373402 A000040 lists the primes, differences A001223.
%Y A373402 A002808 lists the composite numbers, differences A073783.
%Y A373402 A046933 counts composite numbers between primes.
%Y A373402 A065855 counts composite numbers up to n.
%Y A373402 Cf. A006512, A007674, A049093, A068781, A072284, A077641, A174965, A251092, A373198, A373408, A373411.
%K A373402 nonn
%O A373402 1,2
%A A373402 _Gus Wiseman_, Jun 10 2024